The main mathematical models for transportation analysis deal with the determination on how the demand mobility patterns that model in terms of an origin to destination trip matrix is assigned onto a road network under certain conditions to explain the link and paths flows generated by the trips between origins and destinations, consequence of the socioeconomic activities in the región under study. The main modeling hypothesis is base don the principle of Wardrop, formulated in the early 50’s, a behavioral hypothesis on how travelers select the paths in the network under the prevailin traffic conditions, usually congested conditions.

The primary model that translates Wardrop’s principle in mathematical terms is a variational inequalities model which only under specific conditions has an equivalent convex optimization model. Due to the inherent difficulties in dealing with variational inequalities, most of the research work in the 80’s and 90’s addressed the particular case of the convex optimization models. However, the evidence that the underlying hypothesis were rather restrictive and the need of including the temporal dimension in the models in order to appropriately address the inherent time dependencies of traffic phenomena, draw the research interest to search for solutions for the variational inequalities models, namely for the dynamic generalization of Wardrop’s principle to solve numerically the dynamic assignment models whose solutions provide an insight on how traffic flows evolve with time on the network and give raise to congestion.

But dynamic models ask for dynamic inputs, in other words the origin-destination trip matrices modeling traffic patterns must also be time-dependent. This adds a new difficulty as these matrices are directly observable. The time dependent data measurement of traffic variables made available by the new technologies has opened the door to efficient algorithmic approaches based on Kalman Filter.

This paper will present a summary overview of the modeling hypothesis, the variational inequalities models and their alternative, will discuss some of the numerical approaches to solve the models, and the related Kalman Filter approaches to the estimation of time-dependent OD matrices. The paper will conclude discussing some of the numerical results obtained by the CENIT research group.